圆周旋转编码的周期结构与薛定谔算子
Periodic structure and Schrodinger operators for codings of circle rotations
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中文总结 AI 辅助
该研究针对圆周旋转的2区间编码子转移，改进了3-块Gordon结构的判定结果，刻画了几乎必然具有该结构的参数范围，并给出不具有该结构的点集的豪斯多夫维数性质。
中文摘要 AI 辅助
我们考虑对于任意无理数α和区间I⊂𝕋，通过将α重复旋转下的轨道编码为属于I或I^c所得到的2区间编码子转移X^(I,α)。每个序列c∈X^(I,α)都有一个对应的薛定谔算子H_c，文献[kaminaga]中证明，如果α的连分数展开的数字的上极限至少为4，那么几乎所有的c∈X^(I,α)都具有所谓的3-块Gordon结构，这意味着算子H_c没有特征值。我们显著改进了这一结果，完全刻画了几乎必然的3-块Gordon结构，证明实际上除了可数对(α,|I|)之外，所有(α,I)都满足该结构；其中α与银均值是莫比乌斯等价的，且|I|∈ℤα + f(α)，而f(α)是一个特定的无穷级数，取值为1/2、α/2或(α+1)/2。我们还证明，对于具有全测度的α集合，以及每个I，其轨道编码不具有3-块Gordon结构的点集的豪斯多夫维数都远离1。
英文摘要
We consider, for any irrational $α$ and interval $I \subset \mathbb{T}$, the 2-interval coding subshift $X^{(I, α)}$ induced by coding orbits under repeated rotation by $α$ via membership in $I$ or $I^c$. Each sequence $c \in X^{(I, α)}$ has an associated Schrödinger operator $H_c$, and in \cite{kaminaga} it was proved that if the continued fraction of $α$ has digits with limsup at least $4$, then almost every $c \in X^{(I, α)}$ has so-called $3$-block Gordon structure, which implies that the operator $H_c$ has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all $(α,I)$ except for a countable set of pairs $(α, |I|)$ where $α$ is Möbius equivalent to the silver mean and $|I| \in \mathbb{Z}α+ f(α)$ where $f(α)$ is a specific infinite series taking value either $\frac{1}{2}, \fracα{2}$, or $\frac{α+1}{2}$. We also show that for a set of $α$ of full measure, and for every $I$, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.
发表机构
- University of Denver（丹佛大学）
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